Approximation with C1-Cubics: A Counterexample

A smooth bivariate function can, in general, be approximated merely with order $ O(h^3)$ by continuously differentiable piecewise polynomials of total degree $ \le 3$ on regular triangulations with grid width $ h$.

\includegraphics[width=0.6\linewidth,clip]{C1CubicsFig.eps}

The loss of approximation order (local Taylor approximations have order $ 4$) is particularly surprising in view of the existence of compactly supported basis functions.


$ \square$ with C. de Boor, Proc. Amer. Math. Soc. 87 (1983), 649-655.

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